Toric曲面几何造型与半代数样条研究

发布时间：2018-03-31 01:12

本文选题：toric曲面　切入点：toric退化　出处：《大连理工大学》2016年博士论文

【摘要】：参数曲线曲面是计算机辅助几何设计(Computer Aided Geometric Design,简称为CAGD)的重要研究内容.目前,对参数曲线曲面的研究主要集中在对Bezier, B样条与NURBS (Non-Uniform Rational B-Splines)曲线曲面的研究.Toric曲面是有理Bezier曲面的一种多边形推广形式,继承了很多Bezier曲面的造型优点.本文研究toric曲面的几何连续条件和近似极小toric曲面的构造,并将toric曲面应用到数据拟合和管道拼接中.多元样条函数也是几何造型的一个重要工具,常用的研究方法有光滑余因子方法,B网方法,B样条方法,同调方法等.在本文中,我们利用同调代数的方法研究代数曲线剖分下的多元样条函数空间.本文主要工作包括：1.在CAGD中,参数曲面的几何连续是一个非常重要的研究内容.针对toric曲面的几何连续问题,我们推导toric Bernstein基函数的一阶和二阶偏导性质,证明当提升函数满足提升准则时,toric曲面沿边界处的一阶和二阶偏导在toric退化的过程中保持不变,并由此给出toric曲面的一阶几何连续和曲率连续的充要条件.通过指定特殊的提升函数,使得toric曲面沿边界区域退化为张量积型或三角型的有理Bezier曲面.由有理Bezier曲面的一阶几何连续和曲率连续的已知结果,给出toric曲面基于控制结构几何关系的一阶几何连续和曲率连续的充分条件,并给出了一些具体构造的实例.2.极小曲面研究中的一个著名问题是求解Plateau问题,即以给定的边界闭曲线为条件,求解极小曲面.在实际应用中,已知的边界往往是多边的,我们结合toric曲面的参数域是任意凸多边形,由此考虑Plateau-toric问题,使用Dirichlet泛函代替能量泛函求解,得到近似极小toric曲面的一个构造方法,并通过实例验证了方法的可行性.3.拟合数据点集并重构曲面是几何造型中研究的一个重要问题.由于toric曲面是一种多边参数曲面,我们使用toric曲面拟合数据点集并重构曲面,当点集的参数域为凸多边域时,无需对点集的参数域进行剖分,即可得到一个整体拟合的多边参数曲面.其次,构造多管道的过渡曲面在模具设计,工业零部件制造等领域有着广泛应用.借助几何连续条件,我们使用两片toric曲面来构造多管道的过渡曲面.通过这两个应用,可以看出toric曲面不仅保持了有理Bezier曲面构造简单,形状可调等优点,而且参数域为任意的凸多边形,可减少造型中曲面片的个数,避免了多片曲面间的拼接问题.4.几何造型中的一个重要研究对象是多元样条函数,而同调代数是研究多元样条函数的一种有效工具.我们推广线性剖分下的多元样条函数的同调方法到任意代数曲线剖分下的多元样条函数.由Bezout定理可知,2条n次代数曲线最多可相交于n2个点,本文使用同调代数的方法讨论了构成剖分的N条n次代数曲线相交于1个点和n2个点的情况.利用交换代数和toric退化的相关理论分别证明了这两种剖分下的多元样条模空间与线性剖分下的多元样条模空间的关系,分析了这两种剖分下的样条模空间CT(△)的结构,并给出了样条模空间CT(△)的Hilbert多项式与多元样条函数空间CTd(△)的维数公式.
[Abstract]:Parametric curve and surface in computer aided geometric design (Computer Aided Geometric Design, referred to as CAGD) is the important research content. At present, the research of parametric curves and surfaces are mainly focused on Bezier, B and NURBS spline (Non-Uniform Rational B-Splines) on the.Toric surface curve is a polygon generalization of rational Bezier surfaces. Bezier inherits many advantages. The geometric surface modeling of toric surface and the continuity conditions of toric approximate minimal surfaces, and toric surface will be applied to the data fitting and splicing pipeline. A multivariate spline function is an important tool for geometric modeling, the research methods are commonly used smoothing cofactor method, B network method B, spline method, homology method. In this paper, we use multivariate spline space method of algebraic curve section under the homological algebra. In this paper, the main To work includes: 1. in CAGD, the geometric parameters of continuous surface is a very important research content. According to the geometric continuity problem of toric surfaces, we derive the toric Bernstein function one order and two order derivative properties, proved when lifting functions meet the criteria, toric surfaces along the boundary of the first order and two order partial derivatives remain unchanged in the process of toric degradation, and thus gives a second-order geometric surface toric continuous and continuous curvature of sufficient and necessary conditions. By specifying the special lifting function, the toric surface along the boundary area of degradation of rational Bezier surfaces for the tensor product type or triangle. The known results by first order geometric rational Bezier curved continuous and continuous curvature, given toric surface geometric control structure based on the geometric relationship of continuous curvature and sufficient conditions for the continuity, and gives some specific examples of very small structures.2. A famous problem of surface research is to solve the Plateau problem, with the given boundary conditions for closed curve, solving the minimal surface. In practical application, the known boundary is often multilateral, we combine the parameter domain of toric surface is a convex polygon, which account for Plateau-toric, the use of Dirichlet function instead of solving the energy functional and get a method to construct approximate minimal toric surfaces, and proves the feasibility of the method of.3. fitting data points and surface reconstruction is an important research problem in geometric modeling. The toric surface is a multilateral parametric surface, we use toric surface fitting data points and surface reconstruction, when the parameter domain the point set is convex multi domain, the parameter domain without the need for the point set is divided, can get a whole multilateral parametric surface fitting. Secondly, construct the transitional curved pipe In the mold design, parts manufacturing industry has been widely used. By means of geometric continuous conditions, the transition surface we use two pieces of toric surface to construct multiple pipelines. Through the two application, we can see that the toric surface not only keeps the rational Bezier surface has the advantages of simple structure, adjustable shape etc., and the parameter domain is convex a polygon, which can reduce the number of patches in the other, to avoid an important research object in the tiling problem.4. geometric modeling several surfaces between the multivariate spline, and homological algebra is an effective tool to study the multivariate spline function. We generalize homological methods of multivariate spline function for linear cutting under the multiple samples of arbitrary algebraic curves based on the partition function. By Bezout theorem, 2 n algebraic curves can intersect at N2 points, this paper uses the structure of homological algebra is discussed Split N n algebraic curves intersect at 1 points and N2 points. Using the exchange theory of algebra and toric degradation respectively prove the relationship of multivariate spline space model of the two division of the multivariate spline space and linear model based on the partition, and analyzes the two kinds of based on the partition of the spline space model CT (delta) structure, and gives the spline space model CT (delta) Hilbert polynomial and multivariate spline space CTd (delta) dimension formula.

【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TP391.7

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